The dynamic Schr\"odinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a reference process. We propose a novel sampling-based iterative algorithm, the iterated diffusion bridge mixture (IDBM) procedure, aimed at solving the dynamic Schr\"odinger bridge problem. The IDBM procedure exhibits the attractive property of realizing a valid transport between the target probability measures at each iteration. We perform an initial theoretical investigation of the IDBM procedure, establishing its convergence properties. The theoretical findings are complemented by numerical experiments illustrating the competitive performance of the IDBM procedure. Recent advancements in generative modeling employ the time-reversal of a diffusion process to define a generative process that approximately transports a simple distribution to the data distribution. As an alternative, we propose utilizing the first iteration of the IDBM procedure as an approximation-free method for realizing this transport. This approach offers greater flexibility in selecting the generative process dynamics and exhibits accelerated training and superior sample quality over larger discretization intervals. In terms of implementation, the necessary modifications are minimally intrusive, being limited to the training loss definition.

We analyze a general problem in a crowd-sourced setting where one user asks a question (also called item) and other users return answers (also called labels) for this question. Different from existing crowd sourcing work which focuses on finding the most appropriate label for the question (the "truth"), our problem is to determine a ranking of the users based on their ability to answer questions. We call this problem "ability discovery" to emphasize the connection to and duality with the more well-studied problem of "truth discovery". To model items and their labels in a principled way, we draw upon Item Response Theory (IRT) which is the widely accepted theory behind standardized tests such as SAT and GRE. We start from an idealized setting where the relative performance of users is consistent across items and better users choose better fitting labels for each item. We posit that a principled algorithmic solution to our more general problem should solve this ideal setting correctly and observe that the response matrices in this setting obey the Consecutive Ones Property (C1P). While C1P is well understood algorithmically with various discrete algorithms, we devise a novel variant of the HITS algorithm which we call "HITSNDIFFS" (or HND), and prove that it can recover the ideal C1P-permutation in case it exists. Unlike fast combinatorial algorithms for finding the consecutive ones permutation (if it exists), HND also returns an ordering when such a permutation does not exist. Thus it provides a principled heuristic for our problem that is guaranteed to return the correct answer in the ideal setting. Our experiments show that HND produces user rankings with robustly high accuracy compared to state-of-the-art truth discovery methods. We also show that our novel variant of HITS scales better in the number of users than ABH, the only prior spectral C1P reconstruction algorithm.

Many inference scenarios rely on extracting relevant information from known data in order to make future predictions. When the underlying stochastic process satisfies certain assumptions, there is a direct mapping between its exact classical and quantum simulators, with the latter asymptotically using less memory. Here we focus on studying whether such quantum advantage persists when those assumptions are not satisfied, and the model is doomed to have imperfect accuracy. By studying the trade-off between accuracy and memory requirements, we show that quantum models can reach the same accuracy with less memory, or alternatively, better accuracy with the same memory. Finally, we discuss the implications of this result for learning tasks.

While generalized linear mixed models (GLMMs) are a fundamental tool in applied statistics, many specifications -- such as those involving categorical factors with many levels or interaction terms -- can be computationally challenging to estimate due to the need to compute or approximate high-dimensional integrals. Variational inference (VI) methods are a popular way to perform such computations, especially in the Bayesian context. However, naive VI methods can provide unreliable uncertainty quantification. We show that this is indeed the case in the GLMM context, proving that standard VI (i.e. mean-field) dramatically underestimates posterior uncertainty in high-dimensions. We then show how appropriately relaxing the mean-field assumption leads to VI methods whose uncertainty quantification does not deteriorate in high-dimensions, and whose total computational cost scales linearly with the number of parameters and observations. Our theoretical and numerical results focus on GLMMs with Gaussian or binomial likelihoods, and rely on connections to random graph theory to obtain sharp high-dimensional asymptotic analysis. We also provide generic results, which are of independent interest, relating the accuracy of variational inference to the convergence rate of the corresponding coordinate ascent variational inference (CAVI) algorithm for Gaussian targets. Our proposed partially-factorized VI (PF-VI) methodology for GLMMs is implemented in the R package vglmer, see https://github.com/mgoplerud/vglmer . Numerical results with simulated and real data examples illustrate the favourable computation cost versus accuracy trade-off of PF-VI.

Focusing on stochastic programming (SP) with covariate information, this paper proposes an empirical risk minimization (ERM) method embedded within a nonconvex piecewise affine decision rule (PADR), which aims to learn the direct mapping from features to optimal decisions. We establish the nonasymptotic consistency result of our PADR-based ERM model for unconstrained problems and asymptotic consistency result for constrained ones. To solve the nonconvex and nondifferentiable ERM problem, we develop an enhanced stochastic majorization-minimization algorithm and establish the asymptotic convergence to (composite strong) directional stationarity along with complexity analysis. We show that the proposed PADR-based ERM method applies to a broad class of nonconvex SP problems with theoretical consistency guarantees and computational tractability. Our numerical study demonstrates the superior performance of PADR-based ERM methods compared to state-of-the-art approaches under various settings, with significantly lower costs, less computation time, and robustness to feature dimensions and nonlinearity of the underlying dependency.

One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in a poor estimation when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to two typical scenarios for dynamical systems: parameter identification and estimation with an affine structure of the system, and nonlinear parametric approximation without prior knowledge.

In this paper, we introduce and study matching methods based on distance profiles. For the matching of point clouds, the proposed method is easily implementable by solving a linear program, circumventing the computational obstacles of quadratic matching. Also, we propose and analyze a flexible way to execute location-to-location matching using distance profiles. Moreover, we provide a statistical estimation error analysis in the context of location-to-location matching using empirical process theory. Furthermore, we apply our method to a certain model and show its noise stability by characterizing conditions on the noise level for the matching to be successful. Lastly, we demonstrate the performance of the proposed method and compare it with some existing methods using synthetic and real data.

Federated learning (FL) provides a privacy-preserving approach for collaborative training of machine learning models. Given the potential data heterogeneity, it is crucial to select appropriate collaborators for each FL participant (FL-PT) based on data complementarity. Recent studies have addressed this challenge. Similarly, it is imperative to consider the inter-individual relationships among FL-PTs where some FL-PTs engage in competition. Although FL literature has acknowledged the significance of this scenario, practical methods for establishing FL ecosystems remain largely unexplored. In this paper, we extend a principle from the balance theory, namely ``the friend of my enemy is my enemy'', to ensure the absence of conflicting interests within an FL ecosystem. The extended principle and the resulting problem are formulated via graph theory and integer linear programming. A polynomial-time algorithm is proposed to determine the collaborators of each FL-PT. The solution guarantees high scalability, allowing even competing FL-PTs to smoothly join the ecosystem without conflict of interest. The proposed framework jointly considers competition and data heterogeneity. Extensive experiments on real-world and synthetic data demonstrate its efficacy compared to five alternative approaches, and its ability to establish efficient collaboration networks among FL-PTs.

This paper considers learning the hidden causal network of a linear networked dynamical system (NDS) from the time series data at some of its nodes -- partial observability. The dynamics of the NDS are driven by colored noise that generates spurious associations across pairs of nodes, rendering the problem much harder. To address the challenge of noise correlation and partial observability, we assign to each pair of nodes a feature vector computed from the time series data of observed nodes. The feature embedding is engineered to yield structural consistency: there exists an affine hyperplane that consistently partitions the set of features, separating the feature vectors corresponding to connected pairs of nodes from those corresponding to disconnected pairs. The causal inference problem is thus addressed via clustering the designed features. We demonstrate with simple baseline supervised methods the competitive performance of the proposed causal inference mechanism under broad connectivity regimes and noise correlation levels, including a real world network. Further, we devise novel technical guarantees of structural consistency for linear NDS under the considered regime.

Motivated by policy gradient methods in the context of reinforcement learning, we derive the first large deviation rate function for the iterates generated by stochastic gradient descent for possibly non-convex objectives satisfying a Polyak-Lojasiewicz condition. Leveraging the contraction principle from large deviations theory, we illustrate the potential of this result by showing how convergence properties of policy gradient with a softmax parametrization and an entropy regularized objective can be naturally extended to a wide spectrum of other policy parametrizations.